$ A = \left[\begin{array}{rrr}4 & 0 & 2\end{array}\right]$ $ D = \left[\begin{array}{r}1 \\ 1 \\ 2\end{array}\right]$ Is $ A D$ defined?
Answer: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ A$ , have? How many rows does the second matrix, $ D$ , have? Since $ A$ has the same number of columns (3) as $ D$ has rows (3), $ A D$ is defined.